Closed Range and Essential Norms of Composition Operators on Weighted Dirichlet Spaces

@misc{gu2026closed,
  title = {{Closed Range and Essential Norms of Composition Operators on Weighted Dirichlet Spaces}},
  author = {Caixing Gu and Li He and Xiaofeng Wang and Yuanhao Yan},
  howpublished = {arXiv},
  year = {2026},
  month = sep,
  doi = {10.48550/arxiv.2609.23667},
  eprint = {2609.23667},
  archivePrefix = {arXiv},
  url = {https://arxiv.org/abs/2609.23667},
}

TY  - UNPB
TI  - Closed Range and Essential Norms of Composition Operators on Weighted Dirichlet Spaces
AU  - Caixing Gu
AU  - Li He
AU  - Xiaofeng Wang
AU  - Yuanhao Yan
PY  - 2026
DA  - 2026/09/20
PB  - arXiv
DO  - 10.48550/arxiv.2609.23667
UR  - https://arxiv.org/abs/2609.23667
AB  - We study closed range and essential norms of bounded composition operators C_φ on weighted Dirichlet spaces 𝒟_α. For −1<α<0, we first consider maps whose images are obtained by removing a compact subset from a simply connected subdomain of 𝔻. In this setting, closed range is equivalent to the Reverse Carleson property and the Reverse Carleson Disk Condition for the counting measure μ_φ,α, uniform weighted area density of the image, 𝕋⊂φ(𝔻)̅, and the inclusion of an outer annulus in the image. We also obtain a closed range characterization under the Uniform Tail Condition. For all α>−1, we establish two-sided essential norm estimates in terms of local averages of the generalized Nevanlinna counting function and an exact formula using boundary tail integrals. If the normalized counting density has vanishing oscillation at the boundary, we obtain exact formulas in terms of its boundary limsup, Berezin transform, and local averages.
ER  -